Unlike thermal phase transitions driven by temperature, quantum phase transitions occur at T=0 driven by quantum fluctuations. The paradigm example is the transverse-field Ising model: H = −J Σ σziσzi+1 − Γ Σ σxi. As the transverse field Γ increases relative to J, the system transitions from a ferromagnetically ordered phase (spins aligned, Γ/J < 1) to a paramagnetic phase (spins polarized along x, Γ/J > 1), with a quantum critical point at Γ/J = 1.
At the critical point, quantum fluctuations are scale-invariant — the correlation length diverges, and the gap Δ ~ |Γ−J|zν vanishes. The quantum critical point controls a fan-shaped region at finite T: the quantum critical fan. Critical exponents: z=1, ν=1 in 1D (exact via Jordan-Wigner fermionization).